Records

A record is a Sigma type with named projections. The basic syntax looks like this:

\record R (p_1 : A_1) ... (p_n : A_n) {
  | f_1 : B_1
  ...
  | f_k : B_k
}

where f_1, … f_k and p_1, … p_n are fields of R. Note that records do not have parameters and p_1, … p_n are also fields of R. The only difference between p_i and f_j is that f_j are visible in the scope of R and p_i are not:

\func test1 => f_1
\func test2 => R.f_1
\func test3 => R.p_1

-- p_1 is not in scope in the following function:
-- \func test4 => p_1

It is also possible to write \field f_i : B_i instead of | f_i : B_i, but there is a difference between these notations, which is discussed below.

Fields can be accessed using projection functions:

\func test5 (x : R) => R.p_1 {x}
\func test6 (x : R) => f_1 {x}

An alternative way to access fields is provided by the following syntax:

\func test5' (x : R) => x.p_1
\func test6' (x : R) => x.f_1

This syntax is allowed only when the expression before . is a variable with an explicitly specified type which is a record. If x : R and f is a field of R, then x.f is equivalent to R.f {x}.

The type A_i can depend on variables p_1, … p_{i-1}. The type B_i can depend on variables p_1, … p_n and f_1, … f_{i-1}.

Records are essentially Sigma types. For example, the record R above is equivalent to the Sigma type \Sigma (p_1 : A_1) … (p_n : A_n) (f_1 : B_1) … (f_k : B_k).

Instances of type R can be created using new expression. Any of the variants of the syntax listed below can be used, they are all equivalent. See Class extensions for more information about new expressions and related constructions.

\func r1 => \new R a_1 ... a_n { | f_1 => b_1 ... | f_k => b_k }
\func r2 => \new R { | p_1 => a_1 ... | p_n => a_n | f_1 => b_1 ... | f_k => b_k }
\func r3 => \new R a_1 ... a_n b_1 ... b_k
\func r4 => \new R a_1 ... a_i { | p_{i+1} => a_{i+1} ... | p_n => a_n | f_1 => b_1 ... | f_k => b_k }
\func r5 => \new R a_1 ... a_n b_1 ... b_i { f_{i+1} => b_{i+1} ... | f_k => b_k }

The same function can also be defined using copattern matching:

\func r6 : R \cowith
  | p_1 => a_1
  ...
  | p_n => a_n
  | f_1 => b_1
  ...
  | f_k => b_k

Records satisfy the eta rule. This means that the expression \new R r.p_1 … r.p_n r.f_1 … r.f_k is equivalent to r.

Properties

Some fields can be marked as a property. This is done by using the keyword \property instead of \field:

\record NegativeInt {
  \field x : Int
  \property isNeg : x < 0
}

The type of a property must be a proposition, otherwise the definition does not typecheck.

If A is a proposition, then | f : A is also marked as a property. In this case, f can be defined as a normal field, which is not a property, by writing \field f : A.

Properties do not evaluate. Thus, they are related to fields in the same way as lemmas are related to functions. For example, consider the following function:

\func test (x : Int) (p : x < 0) => isNeg {\new NegativeInt x p}

Then test x p evaluates to p if isNeg is not a property and does not evaluate if it is.

Extensions

An extension S of a record R is another record which adds some fields to R and implements some of the fields of R. The record R is called a super class of S and S is called a subclass of R. If R is the definition of a record from the beginning of this page, then an extension S of R can be defined as follows:

\record S (r_1 : D_1) ... (r_t : D_t) \extends R {
  | g_1 : C_1
  ...
  | g_m : C_m
  | p_{i_1} => a_{i_1}
  ...
  | p_{i_q} => a_{i_q}
  | f_{j_1} => b_{j_1}
  ...
  | f_{j_s} => b_{j_s}
}

Here expressions a_i and b_j have types A_i and B_j respectively. Expressions a_i and b_j may refer to any field of S, but implementations must not form a cycle.

The type S is a subtype of R. That is, every expression of type S is also of type R.

A record is equivalent to the Sigma type, consisting of all of its unimplemented fields. For example, consider the following records:

\record C (x y : Nat) {
  | x<=y : x <= y
  | y<=0 : y <= 0
}

\record D \extends C {
  | y => x
  | x<=y => <=-reflexive x
}

Then D is equivalent to \Sigma (x : Nat) (x <= 0):

\func fromD (d : D) : \Sigma (x : Nat) (x <= 0) => (d.x, d.y<=0)
\func toD (p : \Sigma (x : Nat) (x <= 0)) => \new D p.1 p.2
\func fromToD (d : D) : toD (fromD d) = d => idp
\func toFromD (p : \Sigma (x : Nat) (x <= 0)) : fromD (toD p) = p => idp

where idp is the proof by reflexivity. This works since both records and Sigma types satisfy eta rules.

A record can extend several records. If these records extend some base record themselves, then the fields of this base record will not be repeated in the final record. For example, consider the following records:

\record A (x : Nat)
\record B \extends A
\record C \extends A
\record D \extends B,C

Then D has a single field x. If super classes have fields with the same name which are not defined in some common super class, then the final record will have several different fields with the same name. In order to access these fields, fully qualified names should be used:

\record B (x : Nat)
\record C (x : Nat)
\record D \extends B,C

\func fromD (d : D) : \Sigma Nat Nat => (B.x {d}, C.x {d})
\func toD (p : \Sigma Nat Nat) => \new D p.1 p.2
\func fromToD (d : D) : toD (fromD d) = d => idp
\func toFromD (p : \Sigma Nat Nat) : fromD (toD p) = p => idp

This

Every field of a record R has additional implicit parameter of type R, which can be referred to with the keyword \this:

\record R (X : \Type) (t : X -> X)

\func f (r : R) => r.t

\record S \extends R {
  | x : X
  | p : f \this x = x
}

The keyword \this can appear only in arguments of definitions and only in those arguments, which in turn satisfy this condition.

Override

The type of a field or a property can be overridden with a subtype in a subclass using keyword \override:

\record R

\record S \extends R

\record A {
  | f : R
}

\record B \extends A {
  \override f : S
}